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We describe and implement iMapD, a computer-assisted approach for accelerating the exploration of uncharted effective Free Energy Surfaces (FES), and more generally for the extraction of coarse-grained, macroscopic information from atomistic or stochastic (here Molecular Dynamics, MD) simulations. The approach functionally links the MD simulator with nonlinear manifold learning techniques. The added value comes from biasing the simulator towards new, unexplored phase space regions by exploiting the smoothness of the (gradually, as the exploration progresses) revealed intrinsic low-dimensional geometry of the FES.

We link nonlinear manifold learning techniques for data analysis/compression with model reduction techniques for evolution equations with time scale separation. In particular, we demonstrate a `"nonlinear extension" of the POD-Galerkin approach to obtaining reduced dynamic models of dissipative evolution equations. The approach is illustrated through a reaction-diffusion PDE, and the performance of different simulators on the full and the reduced models is compared. We also discuss the relation of this nonlinear extension with the so-called "nonlinear Galerkin" methods developed in the context of Approximate Inertial Manifolds., Comment: 20 pages, 8 figures

Temporal integration of equations possessing continuous symmetries (e.g. systems with translational invariance associated with traveling solutions and scale invariance associated with self-similar solutions) in a ``co-evolving'' frame (i.e. a frame which is co-traveling, co-collapsing or co-exploding with the evolving solution) leads to improved accuracy because of the smaller time derivative in the new spatial frame. The slower time behavior permits the use of {\it projective} and {\it coarse projective} integration with longer projective steps in the computation of the time evolution of partial differential equations and multiscale systems, respectively. These methods are also demonstrated to be effective for systems which only approximately or asymptotically possess continuous symmetries. The ideas of projective integration in a co-evolving frame are illustrated on the one-dimensional, translationally invariant Nagumo partial differential equation (PDE). A corresponding kinetic Monte Carlo model, motivated from the Nagumo kinetics, is used to illustrate the coarse-grained method. A simple, one-dimensional diffusion problem is used to illustrate the scale invariant case. The efficiency of projective integration in the co-evolving frame for both the macroscopic diffusion PDE and for a random-walker particle based model is again demonstrated.

Recent developments in multiscale computation allow the solution of ``coarse equations'' for the expected macroscopic behavior of microscopically/stochastically evolving particle distributions without ever obtaining these coarse equations in closed form. The closure is obtained ``on demand'' through appropriately initialized bursts of microscopic simulation. The effective coupling of microscopic simulators with macrosocopic behavior embodied in this approach requires certain decisions about the nature of the unavailable ``coarse equation''. Such decisions include (a) the determination of the highest spatial derivative active in the equation, (b) whether the coarse equation satisfies certain conservation laws, and (c) whether the coarse dynamics is Hamiltonian. These decisions affect the number and type of boundary conditions as well as the nature of the algorithms employed in good solution practice. In the absence of an explicit formula for the temporal derivative, we propose, implement and validate a simple scheme for deciding these and other similar questions about the coarse equation using only the microscopic simulator. Microscopic simulations under periodic boundary conditions are carried out for appropriately chosen families of random initial conditions; evaluating the sample variance of certain statistics over the simulation ensemble allows us to infer the highest order of spatial derivatives active in the coarse equation. In the same spirit we show how to determine whether a certain coarse conservation law exists or not, and we discuss plausibility tests for the existence of a coarse Hamiltonian or integrability. We argue that such schemes constitute an important part of the equation-free approach to multiscale computation., Comment: 13 pages, 7 figures

We present and discuss a framework for computer-aided multiscale analysis, which enables models at a "fine" (microscopic/stochastic) level of description to perform modeling tasks at a "coarse" (macroscopic, systems) level. These macroscopic modeling tasks, yielding information over long time and large space scales, are accomplished through appropriately initialized calls to the microscopic simulator for only short times and small spatial domains. Our equation-free (EF) approach, when successful, can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form. We discuss how the mathematics-assisted development of a computational superstructure may enable alternative descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks (integration over relatively large time and space scales,"coarse" bifurcation analysis, optimization, and control) directly. In effect, the procedure constitutes a systems identification based, "closure on demand" computational toolkit, bridging microscopic/stochastic simulation with traditional continuum scientific computation and numerical analysis. We illustrate these ideas through examples from chemical kinetics (LB, KMC), rheology (Brownian Dynamics), homogenization and the computation of "coarsely self-similar" solutions, and discuss various features, limitations and potential extensions of the approach.